Inductive coupling is used in a number of wireless energy transfer applications such as charging a cordless electronic toothbrush or hybrid vehicle batteries. In coupled inductors, such as transformers, a source, e.g., primary coil, generates energy as an electromagnetic field, and a sink, e.g., a secondary coil, subtends that field such that the energy passing through the sink is optimized, e.g., is as similar as possible to the energy of the source. To optimize the energy, a distance between the source and the sink should be as small as possible, because over greater distances the induction method is highly ineffective.
In resonant coupling, two resonant electromagnetic objects, i.e., the source and the sink, interact with each other under resonance conditions. The resonant coupling transfers energy from the source to the sink over a mid-range distance, e.g., a fraction of the resonant frequency wavelength. Examples of the resonant coupling system are described in U.S. Patent Applications 2008/0278264 and 2007/0222542.
According to coupled-mode theory, strength of the coupling is represented by a coupling coefficient k. The coupling enhancement is denoted by an increase of an absolute value of the coupling coefficient k. Based on the coupling mode theory, the resonant frequency of the resonant coupling system is partitioned into multiple frequencies. For example, in two objects resonance compiling systems, two resonant frequencies can be observed, named even and odd mode frequencies, due to the coupling effect. The coupling coefficient of two objects resonant system formed by two exactly same resonant structures is calculated by partitioning of the even and odd modes according toκ=π|feven−fodd|.  (1)
It is a challenge to enhance the coupling. For example, to optimize the coupling, resonant objects with a high quality factor Q, which is a ratio of inductive reactance to resistance at a given frequency, are selected. Accordingly, it is desired to optimize wireless energy transfer between the source and the sink.